On Tue, 5 Jul 2005, Michael Walter wrote:
> On 7/5/05, Henning Thielemann <lemming at henning-thielemann.de> wrote:
> > The example, again: If you write some common expression like
> >
> > transpose x * a * x
> >
> > then both the human reader and the compiler don't know whether x is a
> > "true" matrix or if it simulates a column or a row vector.
>> But since a, by definition (question), is a 1xN matrix, what's the
> ambiguity exactly?
If you want this definition then you must also interpret any 1x1 matrix
as a real. That's what I wanted to show with my example, that's the way
MatLab works and why it sucks. Multiplication of reals is commutative,
reals are naturally totally ordered and so on, matrices (including 1x1
matrices) don't have these properties. Since it is sensible to work with
one dimensional vectors it is also sensible to work with 1x1 matrices. But
1x1 matrices of this kind are certainly different from 1x1 matrices
produced by transpose x * a * x. A 1x1 matrix in MatLab can thus mean a
scalar, a row 1-vector, a column 1-vector or a 1x1-matrix. (If you accept
the differences between these terms.) Alternatively you could convert the
expected 1x1-matrix into a real which must be checked at run time.
Theoretically vectors are objects which can be scaled and added and
matrices represent linear operators on vectors. (Operators including
linear operators may build a vector space itself, but that's a different
issue.) Why should we convert each vector into the representation of some
linear operator before doing linear algebra and why should we convert this
representation of a linear operator back to the vector after linear
algebra has happened? Would you load an audio signal into a 1xN-matrix or
into a N-vector? Loading it into a 1xN-matrix forces you to check
dynamically and repeatedly if the matrix has really only row. Btw. I would
also load an image into a vector, but a vector with a two-dimensional
index set.